Hello Justine,
First, let's look at the definitions of the two rules:
Midpoint Rule:
The Midpoint Rule is the approximation $$\int_a^b f(x)\,dx\approx M_n$$, where
$$M_n=\frac{b-a}{n}\left[f\left(\frac{x_0+x_1}{2} \right)+f\left(\frac{x_1+x_2}{2} \right)+\cdots+f\left(\frac{x_{n-1}+x_{n}}{2} \right) \right]$$
Trapezoidal Rule:
The Trapezoidal Rule is the approximation $$\int_a^b f(x)\,dx\approx T_n$$, where
$$T_n=\frac{b-a}{2n}\left[f\left(x_0 \right)+2f\left(x_1 \right)+\cdots+2f\left(x_{n-1} \right)+f\left(x_{n} \right) \right]$$
We can divide the interval $[a,b]$ into $2n$ equally spaced partitions, as so we may then write:
$$M_n=\frac{b-a}{n}\left[f\left(\frac{x_0+x_2}{2} \right)+f\left(\frac{x_2+x_4}{2} \right)+\cdots+f\left(\frac{x_{2n-2}+x_{2n}}{2} \right) \right]$$
And so using the midpoint formula, this becomes:
$$M_n=\frac{b-a}{2n}\left[2f\left(x_1 \right)+2f\left(x_3 \right)+\cdots+2f\left(x_{2n-1} \right) \right]$$
Likewise, the Trapezoidal Rule may now be written:
$$T_n=\frac{b-a}{2n}\left[f\left(x_0 \right)+2f\left(x_2 \right)+\cdots+2f\left(x_{2n-2} \right)+f\left(x_{2n} \right) \right]$$
Multiplying both by $$\frac{1}{2}$$ and adding, we find:
$$\frac{1}{2}\left(T_n+M_n \right)=\frac{b-a}{2(2n)}\left[f\left(x_0 \right)+2f\left(x_1 \right)+\cdots+2f\left(x_{2n-1} \right)+f\left(x_{2n} \right) \right]$$
And using the definition of the Trapezoidal Rule, we find:
$$\frac{1}{2}\left(T_n+M_n \right)=T_{2n}$$
